Markov chain Monte Carlo analysis for the selection of a cell-killing model under high-dose-rate irradiation

Mathematical modeling in radiotherapy for cancer: a comprehensive narrative review

Mathematical modeling has long been a cornerstone of radiotherapy for cancer, guiding treatment prescription, planning, and delivery through versatile applications. As we enter the era of medical big data, where the integration of molecular, imaging, and clinical data at both the tumor and patient levels could promise more precise and personalized cancer treatment, the role of mathematical modeling has become even more critical. This comprehensive narrative review aims to summarize the main applications of mathematical modeling in radiotherapy, bridging the gap between classical models and the latest advancements. The review covers a wide range of applications, including radiobiology, clinical workflows, stereotactic radiosurgery/stereotactic body radiotherapy (SRS/SBRT), spatially fractionated radiotherapy (SFRT), FLASH radiotherapy (FLASH-RT), immune-radiotherapy, and the emerging concept of radiotherapy digital twins. Each of these areas is explored in depth, with a particular focus on how newer trends and innovations are shaping the future of radiation cancer treatment. By examining these diverse applications, this review provides a comprehensive overview of the current state of mathematical modeling in radiotherapy. It also highlights the growing importance of these models in the context of personalized medicine and multi-scale, multi-modal data integration, offering insights into how they can be leveraged to enhance treatment precision and patient outcomes. As radiotherapy continues to evolve, the insights gained from this review will help guide future research and clinical practice, ensuring that mathematical modeling continues to propel innovations in radiation cancer treatment.

Evaluation of relative biological effectiveness for diseases of the circulatory system based on microdosimetry

In the next decade, the International Commission on Radiological Protection (ICRP) will issue the next set of general recommendations, for which evaluation of relative biological effectiveness (RBE) for various types of tissue reactions would be needed. ICRP has recently classified diseases of the circulatory system (DCS) as a tissue reaction, but has not recommended RBE for DCS. We therefore evaluated the mean and uncertainty of RBE for DCS by applying a microdosimetric kinetic model specialized for RBE estimation of tissue reactions. For this purpose, we analyzed several RBE data for DCS determined by past animal experiments and evaluated the radius of the subnuclear domain best fit to each experiment as a single free parameter included in the model. Our analysis suggested that RBE for DCS tends to be lower than that for skin reactions, and their difference was borderline significant due to large variances of the evaluated parameters. We also found that RBE for DCS following mono-energetic neutron irradiation of the human body is much lower than that for skin reactions, particularly at the thermal energy and around 1 MeV. This tendency is considered attributable not only to the intrinsic difference of neutron RBE between skin reactions and DCS but also to the difference in the contributions of secondary γ-rays to the total absorbed doses between their target organs. These findings will help determine RBE by ICRP for preventing tissue reactions.

Cell-cycle dependence on the biological effects of boron neutron capture therapy and its modification by polyvinyl alcohol

Boron neutron capture therapy (BNCT) is a unique radiotherapy of selectively eradicating tumor cells using boron compounds (e.g., 4-borono-L-phenylalanine [BPA]) that are heterogeneously taken up at the cellular level. Such heterogenicity potentially reduces the curative efficiency. However, the effects of temporospatial heterogenicity on cell killing remain unclear. With the technical combination of radiation track detector and biophysical simulations, this study revealed the cell cycle-dependent heterogenicity of BPA uptake and subsequent biological effects of BNCT on HeLa cells expressing fluorescent ubiquitination-based cell cycle indicators, as well as the modification effects of polyvinyl alcohol (PVA). The results showed that the BPA concentration in the S/G2/M phase was higher than that in the G1/S phase and that PVA enhances the biological effects both by improving the uptake and by canceling the heterogenicity. These findings might contribute to a maximization of therapeutic efficacy when BNCT is combined with PVA and/or cell cycle-specific anticancer agents.

The impact of dose rate on responses of human lens epithelial cells to ionizing irradiation

The knowledge on responses of human lens epithelial cells (HLECs) to ionizing radiation exposure is important to understand mechanisms of radiation cataracts that are of concern in the field of radiation protection and radiation therapy. However, biological effects in HLECs following protracted exposure have not yet fully been explored. Here, we investigated the temporal kinetics of γ-H2AX foci as a marker for DNA double-strand breaks (DSBs) and cell survival in HLECs after exposure to photon beams at various dose rates (i.e., 150 kVp X-rays at 1.82, 0.1, and 0.033 Gy/min, and 137Cs γ-rays at 0.00461 Gy/min (27.7 cGy/h) and 0.00081 Gy/min (4.9 cGy/h)), compared to those in human lung fibroblasts (WI-38). In parallel, we quantified the recovery for DSBs and cell survival using a biophysical model. The study revealed that HLECs have a lower DSB repair rate than WI-38 cells. There is no significant impact of dose rate on cell survival in both cell lines in the dose-rate range of 0.033-1.82 Gy/min. In contrast, the experimental residual γ-H2AX foci showed inverse dose rate effects (IDREs) compared to the model prediction, highlighting the importance of the IDREs in evaluating radiation effects on the ocular lens.

Modeling for predicting survival fraction of cells after ultra-high dose rate irradiation

Objective. FLASH radiotherapy (FLASH-RT) with ultra-high dose rate (UHDR) irradiation (i.e. > 40 Gy s-1) spares the function of normal tissues while preserving antitumor efficacy, known as the FLASH effect. The biological effects after conventional dose rate-radiotherapy (CONV-RT) with ≤0.1 Gy s-1have been well modeled by considering microdosimetry and DNA repair processes, meanwhile modeling of radiosensitivities under UHDR irradiation is insufficient. Here, we developed anintegrated microdosimetric-kinetic(IMK)model for UHDR-irradiationenabling the prediction of surviving fraction after UHDR irradiation.Approach.TheIMK model for UHDR-irradiationconsiders the initial DNA damage yields by the modification of indirect effects under UHDR compared to CONV dose rate. The developed model is based on the linear-quadratic (LQ) nature with the dose and dose square coefficients, considering the reduction of DNA damage yields as a function of dose rate.Main results.The estimate by the developed model could successfully reproduce thein vitroexperimental dose-response curve for various cell line types and dose rates.Significance.The developed model would be useful for predicting the biological effects under the UHDR irradiation.

Mathematical model combined with microdosimetric kinetic model for tumor volume calculation in stereotactic body radiation therapy

We proposed a new mathematical model that combines an ordinary differential equation (ODE) and microdosimetric kinetic model (MKM) to predict the tumor-cell lethal effect of Stereotactic body radiation therapy (SBRT) applied to non-small cell lung cancer (NSCLC). The tumor growth volume was calculated by the ODE in the multi-component mathematical model (MCM) for the cell lines NSCLC A549 and NCI-H460 (H460). The prescription doses 48 Gy/4 fr and 54 Gy/3 fr were used in the SBRT, and the effect of the SBRT on tumor cells was evaluated by the MKM. We also evaluated the effects of (1) linear quadratic model (LQM) and the MKM, (2) varying the ratio of active and quiescent tumors for the total tumor volume, and (3) the length of the dose-delivery time per fractionated dose (tinter) on the initial tumor volume. We used the ratio of the tumor volume at 1 day after the end of irradiation to the tumor volume before irradiation to define the radiation effectiveness value (REV). The combination of MKM and MCM significantly reduced REV at 48 Gy/4 fr compared to the combination of LQM and MCM. The ratio of active tumors and the prolonging of tinter affected the decrease in the REV for A549 and H460 cells. We evaluated the tumor volume considering a large fractionated dose and the dose-delivery time by combining the MKM with a mathematical model of tumor growth using an ODE in lung SBRT for NSCLC A549 and H460 cells.

Translational study for stereotactic body radiotherapy against non-small cell lung cancer, including oligometastases, considering cancer stem-like cells enable predicting clinical outcome from in vitro data

BACKGROUND

Curative effects of stereotactic body radiotherapy (SBRT) for non-small cell lung cancer (NSCLC) have been evaluated using various biophysical models. Because such model parameters are empirically determined based on clinical experience, there is a large gap between in vitro and clinical studies. In this study, considering the heterogeneous cell population, we performed a translational study to realize the possible linkage based on a modeling approach.

METHODS

We modeled cell-killing and tumor control probability (TCP) considering two populations: progeny and cancer stem-like cells. The model parameters were determined from in vitro survival data of A549 and EBC-1 cells. Based on the cellular parameters, we predicted TCP and compared it with the corresponding clinical data from 553 patients collected at Hirosaki University Hospital.

RESULTS

Using an all-in-one developed model, the so-called integrated microdosimetric-kinetic (IMK) model, we successfully reproduced both in vitro survival after acute irradiation and the 3-year TCP with various fractionation schemes (6-10 Gy per fraction). From the conventional prediction without considering cancer stem cells (CSCs), this study revealed that radioresistant CSCs play a key role in the linkage between in vitro and clinical outcomes.

CONCLUSIONS

This modeling study provides a possible generalized biophysical model that enables precise estimation of SBRT worldwide.

Tumor radioresistance caused by radiation-induced changes of stem-like cell content and sub-lethal damage repair capability

Cancer stem-like cells (CSCs) within solid tumors exhibit radioresistance, leading to recurrence and distant metastasis after radiotherapy. To experimentally study the characteristics of CSCs, radioresistant cell lines were successfully established using fractionated X-ray irradiation. The fundamental characteristics of CSCs in vitro have been previously reported; however, the relationship between CSC and acquired radioresistance remains uncertain. To efficiently study this relationship, we performed both in vitro experiments and theoretical analysis using a cell-killing model. Four types of human oral squamous carcinoma cell lines, non-radioresistant cell lines (SAS and HSC2), and radioresistant cell lines (SAS-R and HSC2-R), were used to measure the surviving fraction after single-dose irradiation, split-dose irradiation, and multi-fractionated irradiation. The SAS-R and HSC2-R cell lines were more positive for one of the CSC marker aldehyde dehydrogenase activity than the corresponding non-radioresistant cell lines. The theoretical model analysis showed that changes in both the experimental-based ALDH (+) fractions and DNA repair efficiency of ALDH (-) fractions (i.e., sub-lethal damage repair) are required to reproduce the measured cell survival data of non-radioresistant and radioresistant cell lines. These results suggest that the enhanced cell recovery in SAS-R and HSC2-R is important when predicting tumor control probability in radiotherapy to require a long dose-delivery time; in other words, intensity-modulated radiation therapy is ideal. This work provides a precise understanding of the mechanism of radioresistance, which is induced after irradiation of cancer cells.

4-Methylumbelliferone administration enhances radiosensitivity of human fibrosarcoma by intercellular communication

Hyaluronan synthesis inhibitor 4-methylumbelliferone (4-MU) is a candidate of radiosensitizers which enables both anti-tumour and anti-metastasis effects in X-ray therapy. The curative effects under such 4-MU administration have been investigated in vitro; however, the radiosensitizing mechanisms remain unclear. Here, we investigated the radiosensitizing effects under 4-MU treatment from cell experiments and model estimations. We generated experimental surviving fractions of human fibrosarcoma cells (HT1080) after 4-MU treatment combined with X-ray irradiation. Meanwhilst, we also modelled the pharmacological effects of 4-MU treatment and theoretically analyzed the synergetic effects between 4-MU treatment and X-ray irradiation. The results show that the enhancement of cell killing by 4-MU treatment is the greatest in the intermediate dose range of around 4 Gy, which can be reproduced by considering intercellular communication (so called non-targeted effects) through the model analysis. As supposed to be the involvement of intercellular communication in radiosensitization, the oxidative stress level associated with reactive oxygen species (ROS), which leads to DNA damage induction, is significantly higher by the combination of 4-MU treatment and irradiation than only by X-ray irradiation, and the radiosensitization by 4-MU can be suppressed by the ROS inhibitors. These findings suggest that the synergetic effects between 4-MU treatment and irradiation are predominantly attributed to intercellular communication and provide more efficient tumour control than conventional X-ray therapy.

A Model for Estimating Dose-Rate Effects on Cell-Killing of Human Melanoma after Boron Neutron Capture Therapy

Boron neutron capture therapy (BNCT) is a type of radiation therapy for eradicating tumor cells through a ¹⁰B(n,α)⁷Li reaction in the presence of ¹⁰B in cancer cells. When delivering a high absorbed dose to cancer cells using BNCT, both the timeline of ¹⁰B concentrations and the relative long dose-delivery time compared to photon therapy must be considered. Changes in radiosensitivity during such a long dose-delivery time can reduce the probability of tumor control; however, such changes have not yet been evaluated. Here, we propose an improved integrated microdosimetric-kinetic model that accounts for changes in microdosimetric quantities and dose rates depending on the ¹⁰B concentration and investigate the cell recovery (dose-rate effects) of melanoma during BNCT irradiation. The integrated microdosimetric–kinetic model used in this study considers both sub-lethal damage repair and changes in microdosimetric quantities during irradiation. The model, coupled with the Monte Carlo track structure simulation code of the Particle and Heavy Ion Transport code System, shows good agreement with in vitro experimental data for acute exposure to ⁶⁰Co γ-rays, thermal neutrons, and BNCT with ¹⁰B concentrations of 10 ppm. This indicates that microdosimetric quantities are important parameters for predicting dose-response curves for cell survival under BNCT irradiations. Furthermore, the model estimation at the endpoint of the mean activation dose exhibits a reduced impact of cell recovery during BNCT irradiations with high linear energy transfer (LET) compared to ⁶⁰Co γ-rays irradiation with low LET. Throughout this study, we discuss the advantages of BNCT for enhancing the killing of cancer cells with a reduced dose-rate dependency. If the neutron spectrum and the timelines for drug and dose delivery are provided, the present model will make it possible to predict radiosensitivity for more realistic dose-delivery schemes in BNCT irradiations.

A theoretical cell-killing model to evaluate oxygen enhancement ratios at DNA damage and cell survival endpoints in radiation therapy

Radio-resistance induced under low oxygen pressure plays an important role in malignant progression in fractionated radiotherapy. For the general approach to predict cell killing under hypoxia, cell-killing models (e.g. the Linear-Quadratic model) have to be fitted to in vitro experimental survival data for both normoxia and hypoxia to obtain the oxygen enhancement ratio (OER). In such a case, model parameters for every oxygen condition needs to be considered by model-fitting approaches. This is inefficient for fractionated irradiation planning. Here, we present an efficient model for fractionated radiotherapy the integrated microdosimetric-kinetic model including cell-cycle distribution and the OER at DNA double-strand break endpoint (OER_DSB). The cell survival curves described by this model can reproduce the in vitro experimental survival data for both acute and chronic low oxygen concentrations. The OER_DSB used for calculating cell survival agrees well with experimental DSB ratio of normoxia to hypoxia. The important parameters of the model are oxygen pressure and cell-cycle distribution, which enables us to predict cell survival probabilities under chronic hypoxia and chronic anoxia. This work provides biological effective dose (BED) under various oxygen conditions including its uncertainty, which can contribute to creating fractionated regimens for multi-fractionated radiotherapy. If the oxygen concentration in a tumor can be quantified by medical imaging, the present model will make it possible to estimate the cell-killing and BED under hypoxia in more realistic intravital situations.

Track Structure Study for Energy Dependency of Electrons and X-rays on DNA Double-Strand Break Induction

Radiation weighting factor wR for photons and electrons has been defined as unity independently of the energy of the particles. However, the biological effects depend on the incident energies according to in vitro experimental data. In this study, we have quantified the energy concentration along electron tracks in terms of dose-mean lineal energy (yD) on chromosome (micro-meter) and DNA (nano-meter) order scales by Monte Carlo simulations, and evaluated the impact of photon energies on DNA double-strand break (DNA-DSB) induction from an experimental study of irradiated cells. Our simulation result shows that the yD values for diagnostic X-rays (60-250 kVp) are higher than that for therapeutic X-rays (linac 6 MV), which agrees well with the tissue equivalent proportional counter (TEPC) measurements. The relation between the yD values and the numbers of γ-H2AX foci for various photon energy spectra suggests that low energy X-rays induce DNA-DSB more efficiently than higher energy X-rays even at the same absorbed dose (e.g., 1.0 Gy). The relative biological effectiveness based on DNA-DSBs number (RBEDSB) is proportionally enhanced as the yD value increases, demonstrating that the biological impact of the photon irradiation depends on energy concentration along radiation tracks of electrons produced in the bio-tissues. Ultimately, our study implies that the value of wR for photons varies depending on their energies.

Intensity Modulated Radiation Fields Induce Protective Effects and Reduce Importance of Dose-Rate Effects

In advanced radiotherapy, intensity modulated radiation fields and complex dose-delivery are utilized to prescribe higher doses to tumours. Here, we investigated the impact of modulated radiation fields on radio-sensitivity and cell recovery during dose delivery. We generated experimental survival data after single-dose, split-dose and fractionated irradiation in normal human skin fibroblast cells (AGO1522) and human prostate cancer cells (DU145). The dose was delivered to either 50% of the area of a T25 flask containing the cells (half-field) or 100% of the flask (uniform-field). We also modelled the impact of dose-rate effects and intercellular signalling on cell-killing. Applying the model to the survival data, it is found that (i) in-field cell survival under half-field exposure is higher than uniform-field exposure for the same delivered dose; (ii) the importance of sub-lethal damage repair (SLDR) in AGO1522 cells is reduced under half-field exposure; (iii) the yield of initial DNA lesions measured with half-field exposure is smaller than that with uniform-field exposure. These results suggest that increased cell survival under half-field exposure is predominantly attributed not to rescue effects (increased SLDR) but protective effects (reduced induction of initial DNA lesions). In support of these protective effects, the reduced DNA damage leads to modulation of cell-cycle dynamics, i.e., less G₁ arrest 6 h after irradiation. These findings provide a new understanding of the impact of dose-rate effects and protective effects measured after modulated field irradiation.

Analysis of the high-dose-range radioresistance of prostate cancer cells, including cancer stem cells, based on a stochastic model

In radiotherapy, cancer stem cells (CSCs) are well recognized as one of the radioresistant cell types. Even in a small subpopulation, CSCs may have an influence on tumor control probability, represented by cell killing after irradiation. However, the relationship between the percentage content of CSCs and the cell survival dose-response curve has not yet been quantitatively clarified. In this study, we developed a cell-killing model for two cell populations (CSCs and progeny cells) to predict the surviving fractions, and compared it with the conventional linear-quadratic (LQ) model. Three prostate cancer cell lines (DU145, PC3 and LNCaP) were exposed to X-rays at doses ranging from 0 to 10 Gy. After the irradiation, we performed clonogenic survival assays to generate the cell survival curves, and carried out flow-cytometric analyses to estimate the percentage content of CSCs for each cell line. The cell survival curves for DU145 cells and PC3 cells seemed not to follow the conventional LQ model in the high dose range (>8 Gy). However, the outputs of the developed model agreed better with the experimental cell survival curves than those of the LQ model. The percentage content of CSCs predicted by the developed model was almost coincident with the measured percentage content for both DU145 cells and PC3 cells. The experiments and model analyses indicate that a small subpopulation of radioresistant CSCs has lower radiosensitivity in the high-dose range, which may lessen the clinical outcome for patients with prostate cancer after high-dose radiation therapy.

Investigation of dose-rate effects and cell-cycle distribution under protracted exposure to ionizing radiation for various dose-rates

During exposure to ionizing radiation, sub-lethal damage repair (SLDR) competes with DNA damage induction in cultured cells. By virtue of SLDR, cell survival increases with decrease of dose-rate, so-called dose-rate effects (DREs). Here, we focused on a wide dose-rate range and investigated the change of cell-cycle distribution during X-ray protracted exposure and dose-response curves via hybrid analysis with a combination of in vitro experiments and mathematical modelling. In the course of flow-cytometric cell-cycle analysis and clonogenic assays, we found the following responses in CHO-K1 cells: (1) The fraction of cells in S phase gradually increases during 6 h exposure at 3.0 Gy/h, which leads to radio-resistance. (2) Slight cell accumulation in S and G2/M phases is observed after exposure at 6.0 Gy/h for more than 10 hours. This suggests that an increase of SLDR rate for cells in S phase during irradiation may be a reproducible factor to describe changes in the dose-response curve at dose-rates of 3.0 and 6.0 Gy/h. By re-evaluating cell survival for various dose-rates of 0.186-60.0 Gy/h considering experimental-based DNA content and SLDR, it is suggested that the change of S phase fraction during irradiation modulates the dose-response curve and is possibly responsible for some inverse DREs.

Investigation of dose-rate effects and cell-cycle distribution under protracted exposure to ionizing radiation for various dose-rates

During exposure to ionizing radiation, sub-lethal damage repair (SLDR) competes with DNA damage induction in cultured cells. By virtue of SLDR, cell survival increases with decrease of dose-rate, so-called dose-rate effects (DREs). Here, we focused on a wide dose-rate range and investigated the change of cell-cycle distribution during X-ray protracted exposure and dose-response curves via hybrid analysis with a combination of in vitro experiments and mathematical modelling. In the course of flow-cytometric cell-cycle analysis and clonogenic assays, we found the following responses in CHO-K1 cells: (1) The fraction of cells in S phase gradually increases during 6 h exposure at 3.0 Gy/h, which leads to radio-resistance. (2) Slight cell accumulation in S and G₂/M phases is observed after exposure at 6.0 Gy/h for more than 10 hours. This suggests that an increase of SLDR rate for cells in S phase during irradiation may be a reproducible factor to describe changes in the dose-response curve at dose-rates of 3.0 and 6.0 Gy/h. By re-evaluating cell survival for various dose-rates of 0.186-60.0 Gy/h considering experimental-based DNA content and SLDR, it is suggested that the change of S phase fraction during irradiation modulates the dose-response curve and is possibly responsible for some inverse DREs.